The Puzzling Price of Corporate Default Risk

Transcription

1 The Puzzling Price of Corporate Default Risk Leandro Saita November 2, 2005 PRELIMINARY Abstract This paper provides an empirical analysis of the risk-return tradeoff for portfolios of corporate bonds. In line with previous results, I find that there is significant compensation for default risk in general, and in particular for the risk associated with the timing of default (jumpto-default risk). Sharpe ratios of portfolios of just over 100 corporate bonds are up to 10 times larger than those of the S&P500 and those for writing 1-year 20%-out-of-the-money put options on the S&P500. Measures of return-to-skewness and return-to-kurtosis tradeoffs also seem to be more favorable in the corporate bond market than in the stock and options markets. Moreover, for some portfolios, jump-todefault risk premia accounts for 60%-85% of expected excess returns, while jump to default itself accounts for less than 20% of the risk in returns, as measured by higher moments of return. It seems hard to reconcile the high excess expected returns with the relatively low risks, as measured by volatility as well as higher moments of returns on portfolios of corporate bonds and covariation of these returns with systematic risk factors. Stanford Graduate School of Business, 518 Memorial Way, c/o PhD Office, Stanford, CA ( lsaita@stanford.edu). I am very thankful to Darrell Duffie for innumerable discussions, guidance, and support. I want to thank Steve Drucker, Laura Goldberg, Steven Grenadier, Stefan Nagel, Kenneth Singleton, and Ilya Strebulaev for helpful comments and discussions. I also want to thank GFI Group and LombardRisk for generously providing default swap data. The latest version of this paper can be downloaded from lsaita 1

2 1 Introduction This paper analyzes conditional expected returns and risks in corporate credit portfolios, comparing them to stock-market counterparts. After accounting for default risk, spread risk, and correlations of these across issuers, investors receive significantly more compensation for corporate credit risk than for equity-return risk. This is true even after accounting for systematic sources of risk and for aversion to the risk of large losses. There is no obvious explanation for this puzzle. By combining a latent-factor reduced-form model 1 for credit spreads with a model of default probabilities that incorporates macroeconomic covariates as well as firm-specific leverage and volatility variables, the results presented here go beyond those previously available by using measures of risk and expected return that are conditional on current market information. To my knowledge, this paper includes the first portfolio mark-to-market risk model for corporate bonds. I compare Sharpe ratios and other measures of risk-return tradeoff for portfolios of corporate bonds to those of the S&P500 index. I find that with portfolios of corporate bonds issued by roughly 100 firms, for example, investors can obtain significantly higher Sharpe ratios than those achievable in the stock market (standard bond index and default-swap indices include at least 100 names). For holding an equal-weighted portfolio of 5-year zerocoupon corporate bonds on roughly 100 names, investors can attain a Sharpe ratio of 0.9, while the Sharpe ratio for the S&P500 index is about Generous allowances for transactions costs have been included. Sharpe ratios for writing 1-year 20%-out-of-the-money put options on the S&P500 index, whos returns I estimate to have comparable skewness and kurtosis to that of the returns on the equal-weighted portfolio of roughly 100 names, also fall short of those for the portfolio. Sharpe ratios for writing 3- month 20%-out-of-the-money put options on the S&P500 index are, however, comparable to those of the equal-weighted portfolio of corporate bonds. This may suggest that excess expected returns on portfolios of corporate bonds might be explained in part by compensation for systematic risks not captured by the model I estimate. The factors that Fama and French (1993) used to explain stock-market returns are statistically significantly correlated with changes in spreads. The economic significance of the Fama-French factors for explaining expected ex- 1 This model is based on the framework of Lando (1994), but does not belong to the affine family. 2

3 cess returns on corporate bonds is limited, however, accounting for a fraction of 1% to 2% of excess expected return on well-diversified portfolios of corporate bonds. I find that less than 40% of the estimated risk premia in corporate credit portfolios are compensation for mark-to-market risk that is not associated with the event of default. The bulk of the risk premia are compensation for bearing the risk of a default event, after conditioning on the factors that determine spread changes. For example, on an equal-weighted portfolio of year zero-coupon corporate bonds, I find that the excess expected return for a 3-year holding period is 2.5%, of which only 0.7% is compensation for markto-market risk. The remaining 1.8% of excess expected returns compensates investors for bearing default-event risk that is not associated with spread risk. This is labeled jump-to-default risk by Collin-Dufresne, Goldstein, and Hugonnier (2004) and Driessen (2005) among others. Jump-to-default risk premia can be measured as the ratio of short-term risk-neutral default probabilities to short-term actual default probabilities. For example, for Xerox, I estimate the actual one-year default probability on December 2000 at 4.8%, and the risk-neutral one-year default probability at 13%, implying a jump-to-default risk premium of 2.7 = Recent evidence that jump-to-default risk may be heavily priced includes Driessen (2005) and Berndt, Douglas, Duffie, Ferguson, and Schranz (2004). Using corporate bond data, ratings-based default probabilities, and taking taxes and liquidity into account, Driessen (2005) estimates a jump-to-default risk premia of Using default swaps for a different set of firms, and Moodys-KMV expected default frequency measure (EDF), Berndt, Douglas, Duffie, Ferguson, and Schranz (2004) estimate that average jump-to-default risk premia vary between 1.5 and 4, for data from 2001 to I estimate that the cross-sectional average jump-to-default risk premia for the most liquid firms that trade in the default swap market range between 1 and 3.5, consistent with previous findings. Driessen (2005) assumes that a firm s default probability is the average historical default frequency of firms in the same credit rating. In addition to possibly ignoring current firm-specific information, this rules out conditioning on current market conditions, which Kavvathas (2001) and others have shown to be significant. Berndt, Douglas, Duffie, Ferguson, and Schranz (2004) use an EDF estimate of conditional default probabilities that is based on distance to default, a volatility-adjusted measure of leverage, but not on other firm-specific covariates, nor on marketwide conditions. In any case, these two papers do not treat portfolios, and therefore do not estimate the much larger compensation for bearing credit risk that can be achieved through diversification. 3

4 Amato and Remolona (2004) argue that idiosyncratic jump-to-default risk in the corporate bond market may be highly priced because there are not enough liquid bonds to allow investors to significantly diversify jumpto-default risk. They point to the fact that credit indices such as the Dow Jones CDX and Dow Jones itraxx have only 125 names and argue that, even with the diversification provided through portfolios of bonds, there is so much skewness in bond returns that idiosyncratic risk may be difficult to diversify with less than exposure to approximately 500 corporate issuers, for their dataset. I find that there is ample compensation in corporate debt portfolios for skewness and kurtosis, in part because there are indeed significant opportunities for diversification even in moderately sized portfolios (with roughly the number of names in current liquid default swap indices), and in part because of the large compensation for the individual issuer risk. Ratios of excess expected return to skewness and to kurtosis (0.5 and 0.3 respectively) for an equal-weighted portfolio of corporate bonds on roughly 100 names, are at least as large as those of the S&P500 index (0.44 and 0.22 respectively) and of writing 3-month 20%-out-of-the-money put options on the index (0.4 and 0.3 respectively). The remainder of the paper is organized as follows. Section 2 presents the models used for pricing corporate bonds and for computing conditional default probabilities, the latter based on that of Duffie, Saita, and Wang (2005). Section 3 describes the data. Section 4 presents the estimated model. Section 5 presents implications of the estimated model for risk premia and for diversification. Section 6 concludes. 4

5 2 The Model The model is based on empirically estimated models of the joint distribution of corporate default intensities, both actual and risk neutral. The risk-neutral framework is that of a latent-factor reduced-form arbitrage-free model of defaultable bond pricing. The model of actual default probabilities is based on that of Duffie, Saita, and Wang (2005). In this section, I describe these models in detail. 2.1 Pricing Framework The pricing framework is that of a latent-factor reduced-form arbitrage-free model of defaultable bond pricing. We take as given a probability space (Ω, F, P) and an information filtration {F t : t 0} satisfying the usual conditions. 2 Under the absence of arbitrage, no market frictions, and some technical conditions, there exists an equivalent martingale (risk-neutral) measure 3 Q, defined by the property that the price at date t of a security promising some contingent amount X F T at time T t, and paying zero in default, is [ P t = E Q e R ] T t r sds 1 {τ>t } X F t, (1) where τ is the random default time, 1 A is the indicator of an event A, and r is the risk-free short-rate process. For the purposes of this paper, I will assume that τ has a risk-neutral intensity, denoted λ Q (t). That is, a Q-martingale is defined by 1 {τ t} t 0 1 {τ>s} λ Q (s) ds, t 0. Under the usual Cox-process (doubly stochastic) assumption, the F t - conditional risk-neutral probability of survival to T is p t = E [ ] [ Q 1 {τ>t } Ft = 1{τ>t} E Q e R ] T λ t Q (s) ds F t. (2) For computational simplicity, I assume that the risk-neutral intensities of default are Q-independent of interest rates. The default-swap pricing implications of any such Q-correlation, for conventional U.S. interest-rate 2 For technical definitions, see Protter (2003). 3 See Harrison and Kreps (1979) for the foundations of this approach, and Delbaen and Schachermayer (1999) for technical conditions. 5

6 behavior, is moderate (Duffie and Singleton 2003). In this case, from (1), the price at time t of a risky zero-coupon bond paying X = 1 at maturity T and nothing in default is P t = δ(t, T ) p t, where δ(t, T ) is the price at time t of a risk-free zero coupon bond with maturity T and a principal of one. ) For each issuer i {1,..., n}, I assume that Xt i = ln (λ Qi (t) is an Ornstein-Uhlenbeck process under P, in that dx i t = ki p (θi p Xi t ) dt + σi dz i,p t, (3) where Z i,p is standard Brownian motion under P, and where kp i, θi p, and σi are constants. I also assume that dx i t = k i q (θ i q X i t) dt + σ i dz i,q t, for Z i,q a Brownian motion under Q. This implies that meaning that ( k dz i,q i q θq i t = ki p θi p σ i ξ i t = ki q θi q ki p θi p σ i + ki q ) ki p X i σ i t dt + dz i,p t, + ki q ki p X i σ i t is the market price of risk associated with Z i,p, which is the risk associated with changes in risk-neutral intensities. That is, ξt i is the mark-to-market risk premia, compensation for the risk of adverse changes in spreads. Following Berndt, Douglas, Duffie, Ferguson, and Schranz (2004), I assume a risk-neutral mean loss given default of 75% of the face value of a corporate bond. Given the estimates of recovery rates for senior unsecured debt from Altman, Resti, and Sironi (2001), I assume an actual mean loss given default of 57%. Further, I assume that loss given default is indepentent (under both P and Q) of Q-intensities ans P-intensities. These assumptions regarding loss given default may have a significant qualitative effect on my results on default risk premia and on the diversification for portfolios of corporate bonds. In particular, positive cross-sectional loss-given-default P-correlation would reduce the high Sharpe ratios, presented in Section 5, for portfolios of corporate bonds. Unfortunately there is not much guidance in the literature on cross-sectional loss-given-default correlation. 6

7 Frye (2000a), Carey and Gordy (2003), Hu and Perraudin (2002), and Altman, Brady, Resti, and Sironi (2006) provide some evidence that actual loss given default and default intensities might be positively correlated, another potential mis-specification of my model. To my knowledge, there is no guidance in the literature regarding correlation between risk-neutral intensities and risk-neutral loss given default. If loss given default risk is systematic, then we should expect risk-neutral mean loss given default to be higher than actual mean loss given default, reflecting risk premia for systematic loss given default risk. Some evidence that loss given default contains systematic risk is provided by Schuerman (2004), Frye (2000a), Frye (2000b), and Altman, Brady, Resti, and Sironi (2006) but there is no evidence in the literature on the size of this possible lossgiven-default risk premia. In order to account for this possible risk premia I have assumed that the risk-neutral mean loss given default is 75%, which is higher than the assumed actual mean loss given default of 57%. A different assumption on risk-neutral loss given default should not significantly change the main results of this paper on portfolio risk and return, but would affect the estimated jump-to-default risk premia. I estimate (issuer by issuer) k i p, θ i p, σ i, k i q, and θ i q, using only the time series of 1-year, 3-year, and 5-year CDS rates for each issuer. I assume that 5-year CDS rates are observed without errors, but that 1-year and 3-year CDS rates are observed with pricing errors, meaning that where C i,m t C i,m t = Ĉi,m t + ν i,m t, is the observed m-year CDS rate for firm i at time t, Ĉ i,m t is is a normally the corresponding model-implied m-year CDS rate, and ν i,m t distributed pricing error with mean zero and standard deviation σm. i The pricing errors are assumed to be serially and cross-sectionally, independent. Issuer-by-issuer maximum-likelihood estimation gives us consistent, although not efficient with respect to the full dataset, 4 estimates for the Q- and P-parameters of the Q-intensity. From the observed market CDS rates, we can then back out fitted values of λ Q i (t), for 0 t T and for 1 i n. For details see Appendix A. 2.2 P-Intensities The model of actual default probabilities is based on that of Duffie, Saita, and Wang (2005), which provides maximum-likelihood estimates of term struc- 4 Fully efficient estimates are given by maximization of the likelihood of the complete dataset, which for computational reasons, is difficult for this setup. 7

8 tures of conditional probabilities of corporate default, incorporating an estimated time series model of firm-specific and macroeconomic covariates. It is assumed that, for firm i, the P-intensity λ P i (t) of default is a function of firm-specific covariates and macroeconomic covariates. The firm-specific covariates are distance-to-default, 5 which is a volatility-adjusted measure of leverage, and (motivated by Shumway (2001)), trailing one-year stock return. For macroeconomic covariates, I use 3-month treasury rates, a simplification of Duffie, Saita, and Wang (2005). Let M t denote the macroeconomic covariates at time t, and St i be the vector of firm-specific covariates for firm i at time t. I assume that ln ( λ P i (t)) = c + θ S i t + γ M t, (4) for parameters c, θ, and γ estimated by maximum likelihood, as explained in Duffie, Saita, and Wang (2005). As mentioned in the introduction, jump-to-default risk premia is measured as the ratio of short-term risk-neutral default probabilities to shortterm actual default probabilities. Technically, for a given firm i, its jump-todefault risk premium as of time t is 2.3 Time Series µ i (t) = λq i (t). (5) λ P i (t) In order to analyze the conditional expectation and standard deviation of returns for portfolios of bonds, we must specify and estimate the joint time series of λ Q i (t), λp i (t), λ Q j (t), and λp j (t), for all firms i and j. In this subsection, all probabilistic properties are under P. I assume that Z i,p t and Z j,p t have, for all (i, j) a common correlation ρ to be estimated. In particular, I assume that Z i = 1 ρ B i + ρ U, (6) 5 The definition of distance to default that I use as a covariate in the actual default intensities is that of Vassalou and Xing (2004) and Duffie, Saita, and Wang (2005), and for a given firm is defined by D t = ln ( At L t ) + ( µ A 1 2 σ2 A) T σ A T, where T is one year, A t is total assets, L t is short-term debt plus one-half of long term debt, 6 and µ A and σ A measure the firm s mean rate of asset growth and asset volatility, respectively. Total assets A t and asset volatility σ A are estimated on the basis of the Black-Scholes option pricing formula, as in Merton (1974). For details, see Appendix B. 8

9 where Bt 1,..., Bn t, and U t are independent standard Brownian Motions. For a given firm i, the distance to default Dt i i and the log asset size Vt are assumed to form a vector Gaussian autoregressive process. In particular [ ] [ ] [ ] ([ ] [ ]) D i t+1 D i Vt+1 i = t kd 0 θ i Vt i + D D i 0 k v θv i t Vt i + ηt+1 i, (7) where [ ηt i Z i,p t = A ] Z i,p t 1 yt i + Bw t, (8) and where yt i is a 2 1 vector of firm-specific shocks to Di t and V t i, w t is a 2 1 vector of market-wide shocks to distances-to-default and log-assets, and the time step is monthly. Here, A is a 2 3 matrix, and B is a 2 2 matrix, standardized by taking A(1, 3) = B(1, 2) = 0. I assume that w 1,..., w T, and y1, 1..., yt 1,..., yn 1,..., yt n are independent standard-normal 2 1 vectors. The time-series behavior of one-year trailing same-firm returns is implied by the time-series model of the firm s distance-to-default and log-asset processes, together with the equations determining distance-to-default and log-assets from leverage and equity, equations (21) and (22), respectively. The macroeconomic covariate driving actual default intensities is the 3- month Treasury-bill rate r 1t. I assume that r t = (r 1t, r 2t ), where r 2t is the 10-year Treasury note rate, satisfies r t+1 = r t + k r (θ r r t ) + C 1/2 r ɛ t+1, (9) where ɛ t are serially independent standard-normal innovations, and the time step is monthly (as before). The coefficients k r, θ r, and C r, are estimated by maximum likelihood, and reported in Duffie, Saita, and Wang (2005). In order to estimate the portion of conditional excess expected returns of bonds and portfolios of bonds that is compensation for stock-market risk factors (see Section 5.3), I use the Fama-French risk factors. I assume that the time-series behavior, for monthly time steps, of the Fama-French factors, F t, is given by F t+1 = F t + k F (θ F F t ) + Σ 1/2 F H t+1, (10) where F t is a 3 1 vector consisting of the time-t returns of the S&P500 index, the SMB, and the HML Fama-French factors respectively, and {H 1,..., H t } are serialy independent 3-dimensional standard-normal random variables. I assume that H t is joint normal with U t and w t, and fix [ ] ] Σ = E P t H t+1 [ Ut+1 U t w t+1 9. (11)

10 I also assume that H t is independent of U s+1 U s and w s for s t. The parameters of (10) are estimated by maximum likelihood. The matrix of covariances between Fama-French factor innovations, and innovations in distance-to-default and log-assets, Σ, is estimated in a second step by equating the corresponding sample moments, as detailed in Appendix B. 2.4 Joint P-distribution of Defaults The probability of survival under P, again assuming doubly-stochastic default, is [ P (τ i > T F t ) = 1 {τi >t}e P e R ] T t λ P i (s) ds Ft. (12) The maximum likelihood estimate of this survival probability is that associated with the estimated time-series model of (S i t, M t), and the estimated intensity parameters (c, θ, γ). Letting p i = P (τ i > T F t ), and p ij = P (τ i > T, τ j > T F t ), the F t - conditional correlation of the indicators of default by firms i and j i within T years, for firms i and j not already in default, are p ij p i p j pi (1 p i ) p j (1 p j ), (13) where p ij is the joint survival probability E P [ e R T t (λ P i (s)+λp j (s)) ds Ft ], (14) which is again, estimated by substituting estimated parameters for their model counterparts. 10

11 3 The Data This section provides a description of the data used in this paper. Data for default probability estimates are those of Duffie, Saita, and Wang (2005). In particular, Moodys Default Risk Service data is used for default times and CRSP/COMPUSTAT is used for all other firm-specific financial data. For the estimation of the bond pricing model of Section 2.1, I use default swap data provided by GFI and Lombard Risk. A default swap, or credit default swap (CDS), is an over-the-counter derivative security designed to transfer credit risk. The buyer of protection pays quarterly insurance premiums, until the expiration of the contract or until a contractually defined credit event, whichever is earlier. If the credit event occurs before the expiration of the default swap, the buyer of protection receives from the seller of protection the difference between the face value and the market value of the underlying debt, less the default-swap premium that has accrued since the last default-swap payment date. The CDS rate is the annualized premium rate, as a fraction of notional. The CDS data for this paper were generously provided by GFI and by Lombard Risk. The GFI database consists of intra-day bid and ask trades and quotes for 1,804 worldwide issuers, including corporates, sovereigns, and government agencies, starting in June 1998 and ending in June Maturities range from under one year to over ten years, the 5-year maturity being the most common. The Lombard Risk CDS data include mid-bid-ask quotes for maturities of 1, 3, 5, and 10 years for 2,714 corporate and sovereign issuers, starting in July 1999 and ending in June I use weekly observations of 1-year, 3-year, and 5-year corporate CDS rates obtained by merging the GFI and Lombard Risk databases. For each issuer, and each week of the sample period, I compute the average of the daily bids, calling this the weekly bid, and similarly for the ask. I define the weekly rate to be the average of the computed weekly bid and weekly ask. For an issuer to be included in my sample, it must have at least 100 weekly observations of its 5-year CDS rate from the merged GFI-Lombard Risk database. Because Duffie, Saita, and Wang (2005) rely only on nonfinancial firms (the Industrial Sector, as defined by Moodys Investor Service), so does this study. Given that the four databases, GFI, Lombard Risk, Moodys Investor Service, and Compustat do not share a common identifier, the databases were hand-matched by issuer name. As a result of this data selection, the CDS dataset for this paper covers 11

12 314 issuers with an average of 109, 127, and 154 weekly observations per issuer, for maturities of 1, 3, and 5 years, respectively. Of these 314 firms, 80.2% are North American, 15.9% are European, 2.9% are Asian, and 1% are Australian. The estimation of the pricing model of Section 2.1, described in detail in Appendix A, is performed for these 314 issuers. The CDS market was not as liquid nor deep in its first few years as it is now. In order to have an homogeneous (over time) set of names for the diversification results presented in Section 5, I further restrict my issuer selection to those issuers that have, for the full period from January 2001 to December 2004, at least two 5-year CDS rate observations for each month. This reduces my sample to 118 names, of which 109 are North American, 8 are European, and 1 is Australian. Table 1 provides sumary statistics of Moodys credit ratings for the issuers in this and the extended datasets. Rating 118-issuer sample 314-issuer sample Aaa 0 3 Aa1 0 2 Aa2 1 6 Aa A1 2 9 A A Baa Baa Baa Ba Ba Ba B1 3 4 B2 2 5 B3 5 6 WR 4 26 Table 1: Moodys rating s distribution of issuers for the two samples. For the bond market, which is less liquid than the CDS market, Driessen (2005) estimates that liquidity accounts on average for 20.6 basis points (bps) of yield spreads. Using the interest-rate swap curve to imply risk-free rates, Longstaff, Mithal, and Neis (2005) report that the non-default component accounts for 5%-10% of corporate bond yield spreads. Blanco, Brennan, and 12

13 Marsh (2004) report that the discrepancy between CDS rates and corporate bond yield spreads is, on average, between 11.6 basis points and 22.5 basis points. There is no clear guidance in the literature on the impact of liquidity on CDS rates, only on bond yield spreads. Here, in order to conservatively account for possible liquidity premia and transaction costs, and given that I will consider portfolios of corporate bonds with positive bond positions only, the default swap rates used in my estimation are 80% of the observed market CDS rates, less 10 basis points. Bid-ask spreads in the CDS market are usually much narrower than 20% of the corresponding rate. Thus I implicitly assume that 20% of the CDS rate plus 10 basis points covers both transaction costs and liquidity premia. While these are perhaps overly generous allowances for the non-credit-related components of observed CDS rates, that would mean that my conclusions regarding the large size of default risk premia are conservative in this regard. 13

14 4 Parameter Estimates In this section, I summarize the results 7 for the pricing model of Section 2.1, the default probability model of Duffie, Saita, and Wang (2005), and the parameter estimates for the time-series model of Section Pricing Parameters Table 2 presents summary statistics for the parameter estimates of the pricing model for the full set of 314 firms. These results are obtained by maximum likelihood estimation, as detailed in Appendix A, imposing the aditional restrictions that k q = k p, (15) θ q θ p. (16) This is a simple way, among others, to impose a non-negative risk premium for adverse (upward) changes in risk-neutral intensity. A non-negative risk premium for changes in risk-neutral intensity is economically intuitive because one can expect corporate bond spreads to widen when the economy declines, 8 and hence investors should be compensated for holding this systematic risk. Percentile k p = k q k q θ q k p θ p σ Table 2: Summary statistics of estimated pricing parameters. The estimates of risk-neutral intensity volatility, σ, presented in Table 2 are consistent with those estimated by Berndt, Douglas, Duffie, Ferguson, and Schranz (2004), who use a pricing model similar to the one I estimate 7 The full set of results for the pricing model for the 314 firms is in Appendix A. 8 See, for example, Fama and French (1989). 14

15 here. The estimates of actual and risk-neutral mean-reversion, k p and k q, presented here seem to be smaller than those of Berndt, Douglas, Duffie, Ferguson, and Schranz (2004), possibly due to the restriction given by equation (16) that I impose on these parameters. In some cases, as shown by the first column of Table 2, the estimated mean-reversion parameter of risk-neutral log-intensity k p is negative. There is no obvious reason to impose a non-negativity constraint on k p, as this is not the mean-reversion parameter of an actual physical stochastic process, but rather, it is the mean-reversion parameter of a process which, possibly, includes a risk premia. Columns 1 and 2 of Table 3 present summary statistics for the estimated standard deviations, σ 1 and σ 3, of the pricing errors for the 1-year and 3-year CDS rates, respectively (CDS rates are measured in basis points). Columns 3 and 4 of Table 3 present summary statistics for the pricing errors as a percentage of the same-firm average 1-year and 3-year CDS rates, respectively. The percentage pricing errors for 1-year and 3-year rates are defined to be the median absolute pricing errors, as a fraction of the corresponding 1-year and 3-year rates, respectively. Volatility Error (%) Percentile σ 1 σ 3 1-year 3-year Table 3: Assesment of goodness of fit of the pricing model. Columns 1 and 2 present summary statistics for the estimated standard deviations σ 1 and σ 3, of pricing errors for the 1-year and 3-year CDS rates. Columns 3 and 4 present the corresponding median absolute pricing errors as percentages of the corresponding rate. Overall, the pricing model of Section 2.1 fits the CDS data relatively well, given that it is just a one-factor model. The 3-year CDS percentage pricing errors have a median mean absolute deviation of 8.4%. As one might expect, given that we are fitting only the 5-year rates exactly, the model has larger percentage pricing errors for the 1-year rates. This could be an indication that a second factor is needed to fit well the full term structure of CDS rates. 15

16 4.2 Actual Default Risk Parameters The model of actual default probabilities is based on that of Duffie, Saita, and Wang (2005), who provide maximum-likelihood estimates of term structures of conditional probabilities of corporate default, incorporating an estimated time series model of firm-specific and macroeconomic covariates. Table 4 presents the estimates of the coefficients determining the dependence of λ P on the covariates, estimated by maximum likelihood as detailed in Duffie, Saita, and Wang (2005), for these covariates. constant distance to default same-firm return 3-month treasury rate (0.256) (0.061) (0.129) (0.035) Table 4: Estimates for the default intensity coefficients on the covariates. Standard errors are presented in parenthesis. 4.3 Time-Series Parameters In order to estimate risk premia on bond-portfolio trading strategies, one must specify the joint conditional distributions under P of λ Q i, λp i, λ Q j, and λ P j, for all firms i and j, as well as the Q-distribution of λq i. This joint distribution is specified by the pricing model of Section 2.1 jointly with the time-series model of covariates driving actual default intensities presented in Section 2.3. Here I present the parameter estimates for the time-series model of the covariates driving actual default intensities. Table 5 presents maximum likelihood estimates for the parameters of the joint time series model for the full set of 314 firms and for monthly time steps. Some observations are in place. First, all parameters shown in Table 5 are estimated to be statistically significant. The mean-reversion parameters of distance to default, k d, and log-assets, k v, are estimated to be positive, and imply an expected half-life of shocks of approximately 20 months for distance to default and 60 months for log-assets. Significant cross-sectional correlation of innovations in risk-neutral logintensities is implied by the correlation estimate, ˆρ = Ignoring liquidity and tax issues, this is consistent with a large common factor driving changes in bond spreads, as suggested by Collin-Dufresne, Goldstein, and Martin (2001). 16

17 parameter estimate standard errors k d k v ρ A(1, 1) A(2, 1) A(1, 2) A(2, 2) A(2, 3) B(1, 1) B(2, 1) B(2, 2) Table 5: Maximum likelihood parameter estimates for the joint time series of distance-to-default, log-assets, and Q-intensity innovations for the full set of 314 firms for a monthly time-step. It is important to note that the estimates of A(1, 1) and A(2, 1) are negative, implying that changes in risk-neutral intensities are negatively correlated with changes in distance to default and log-assets. This, together with the actual default intensity parameter estimates presented in Table 4, is in line with economic intuition: Actual and risk-neutral intensities are, intuitively, positively correlated. That is, as the credit quality of a given firm deteriorates, we expect its credit spreads to widen. The estimates of A(1, 2) and A(2, 2) have the same sign, implying that, as one might expect, for a given firm, distance to default and log-assets are positively correlated. The parameter estimates of the matrix B indicate that there is also correlation in distance to default and log-assets across firms. This is an important determinant of default correlation. The time-series model of covariates driving actual default intensities used in this paper is similar to that of Duffie, Saita, and Wang (2005), but given the need for CDS data on each issuer I consider, I use a much more restricted dataset, as detailed in Section 3, for the estimation. Still, the mean-reversion estimates for distance to default and log-assets presented here are comparable to those of Duffie, Saita, and Wang (2005). The covariance matrix of innovations in distance to default and log-assets, A A + B B, is also comparable to that estimated by Duffie, Saita, and Wang (2005). Given the estimated model of actual default probablities, including the time-series model for the covariates, one can calculate default correlations 17

18 at any given time horizon, for any pair of firms, using (12), (13), and (14). Unfortunately, the resulting default correlation estimates are an order of magnitude smaller than direct sample correlations of default reported by de Servigny and Renault (2002). This is possibly due, for example, to missing covariates in the model of default probabilities, or to measurement noise in the covariates used. In order to study the return risk of portfolios of corporate bonds, it is essential to reasonably capture default correlation. To do this, I propose an adjustment of the time-series parameter matrices A and B that leaves unchaged the marginal distribution of defaults, while increasing default correlation to levels comparable to those implied by the non-parametric method used by de Servigny and Renault (2002). 9 Increasing the modeled default correlations in this way is conservative, in the sense that it weakens the conclusions of the paper regarding the large size of default risk premia for portfolios. See Appendix C for more details. The parameter estimates and standard errors for the time-series model of interest rates and Fama-French factors are provided in Appendix B, Tables 9 and 10. The estimates of Σ [ ( )] = E P t Ht+1 Ut+1 U t, wt+1 that are provided in Table 11 and replicated below (with standard errors in parenthesis), deserve some comment. The estimate is (0.0116) (0.0213) (0.0252) Σ = (0.0111) (0.0209) (0.0254). (17) (0.0120) (0.0239) (0.0294) The elements of the first column of this matrix are negative and statistically significant, implying that innovations in the Fama-French factors, H t, are negatively correlated with innovations in U t, the common component driving default swap rates. This is related to the results of Collin-Dufresne, Goldstein, and Martin (2001), who find that the Fama-French factors are statistically significant for explaining corporate bond spread changes. 10 Consistent with my results here, Collin-Dufresne, Goldstein, and Martin (2001) find that, after accounting for firm-specific and other macroeconomic covariates, changes in corporate bond spreads are negatively correlated with the Fama-French factors. The first column of Σ is also consistent with the results of Elton, Gruber, Agrawal, and Mann (2001) and Schaefer and Strebulaev (2003), who find 9 See Appendix C for details. 10 Ignoring taxes and liquidity issues, default swap rates are equivalent to corporate bond spreads. 18

19 that after accounting for firm-specific and other macroeconomic covariates, corporate bond excess returns are positively correlated with the Fama-French factors. This is consistent with my findings and those of Collin-Dufresne, Goldstein, and Martin (2001), because, for short time periods, after hedging for risk-free interest rates, corporate bond returns are close to proportional to spread changes, with a negative proportionality coefficient. The elements of the second column of Σ are positive, implying that w t,1, the common component of innovations in distance to default, is estimated to be positively correlated with innovations in the Fama-French factors. Intuition for the third column of Σ is more dificult because it relates to w t,2, which affects log-assets jointly with w t,1. 19

20 5 Default Risk Premia I next present estimates of default-risk premia in general, and in particular of jump-to-default risk premia, which is the premia for the risk associated with the timing of default, and of several measures of the risk-return tradeoff in the CDS market. I also present estimates of the part of expected excess returns on corporate bonds that is compensation for systematic stock-market risk factors. I analyze the degree of diversification achievable in the CDS market and how much of the higher moments of returns on a diversified portfolio of corporate bonds can be attributed to jump-to-default risk. 5.1 The Price of Default Risk This subsection presents results on the degree to which default risk, and in particular jump-to-default risk, is priced in the corporate credit market. Consistent with previous findings of Berndt, Douglas, Duffie, Ferguson, and Schranz (2004) and Driessen (2005), average annual risk-neutral default probabilities vary over time, being between 1 and 3.5 times actual default probabilities. Risk-neutral default intensities are, at the median, 3.7 times larger than actual default intensities. For various time horizons of length t, Figure 1 plots the ratio of the annualized conditional risk-neutral default probability to the actual default probability, defined as 1 Q(t) 1/t 1 P (t) 1/t, where Q(t) and P (t) are the cross-sectional average Q- and P-conditional probabilities of survival for t years. Table 6 presents percentiles of the crosssectional distribution of the issuer-by-issuer across-time median annualized P- and Q-default probabilities, and their ratio. Table 7 presents the same statistics as those of Table 6, when considering only the subset of issuers whose estimated median P-intensity is at least 1 basis point, illustrating that the highest percentile of the ratio of risk-neutral to actual default probabilities is due to the safest firms. The assumed mean loss-given-default under P does not affect the estimate of actual default probability, while estimated risk-neutral default probabilities are approximately inversely proportional to the assumed risk-neutral mean loss given default. Even when assuming, as here, that risk-neutral mean loss-given-default is about 50% larger than mean loss-given-default 20

21 (1 Q(t) 1/t )/(1 P(t) 1/t ) Holding Period Date Figure 1: Historical, annualized Q-default probability to P-default probability ratio. percentiles P (1) Q(1) Q(1) 1 P (1) P (5) 1/ Q(5) 1/ Q(5) 1/5 1 P (5) 1/ Table 6: Percentiles of the empirical cross-sectional distribution of the issuerby-issuer across-time median of annualized P- and Q-default probabilities in basis points, and their ratio, for 1-year and 5-year horizons. under the physical measure, risk-neutral annual default probabilities are still about two times larger, at the median, than actual default probabilities. Table 8 illustrates the degree to which jump-to-default risk is priced by presenting percentiles for the distributions of the issuer by issuer across-time 21

22 percentiles P (1) Q(1) Q(1) 1 P (1) P (5) 1/ Q(5) 1/ Q(5) 1/5 1 P (5) 1/ Table 7: Percentiles of the empirical cross-sectional distribution of the issuerby-issuer across-time median of annualized P- and Q-default probabilities in basis points, and their ratio, for 1-year and 5-year horizons, when considering only the subset of issuers whose estimated median P-intensity is at least 1 basis point. medians of λ P, λ Q, and their ratio, for 1-year and 5-year horizons. Jumpto-default risk premia is estimated to be 3.7 at the median, with significant cross-sectional variation. percentiles λ P λ Q λ Q λ P Table 8: Percentiles of the cross-sectional distribution of the issuer-by-issuer across-time-median of λ P and λ Q in basis points per year, and their ratio, for 1-year and 5-year horizons. The results of this section indicate that there is significant default risk premia in general, and in particular, that there is significant jump-to-default risk premia. Risk-neutral expected losses are, at the median, approximately 4 times larger than actual expected losses, as indicated by a ratio of riskneutral to actual default probabilities of 2.94, and assumed risk-neutral and actual mean loss given default of 75% and 57%, respectively. 22

23 5.2 The Risk-Return Tradeoff and Portfolio Formation For any given asset, or portfolio of assets, the Sharpe ratio is the excess expected return per unit of risk, where risk is measured by the standard deviation of returns. Amato and Remolona (2004) argue that in credit markets, higher moments of returns could be more relevant than the second moment for determining the level of risk. Here, in order to analyze the risk-return tradeoff in the corporate credit market, I use a generalization of the Sharpe ratio risk-return measure, [ ] E P P T r(t,t ) (T t) Ft Pt e S(t, T ; n) =, (18) m(t, T ; n) 1/n where P t is the price at time t of the asset in question, r(t, T ) is the continuously compounded annual risk-free rate at time t for year T, and [( [ ]) m(t, T ; n) = E P PT E P PT n ] P t F t F t is the t-conditional n-th central moment of returns for the asset in question. An annualized version of S(t, T ; n) is s(t, T ; n) = P t S(t, T ; n) (T t) 1 1/n. For the special case n = 2, S(t, T ; n) and s(t, T ; n) are the Sharpe ratio and annualized Sharpe ratio, respectively. Equation (18) is a generalization of the Sharpe ratio, which I denote n-th moment Sharpe ratio, because it measures the expected return per unit of risk, where risk is measured by the n-th central moment of returns. Figure 2 presents average (across issuers and across time) annualized n-th moment Sharpe ratios, for holding periods of up to five years, for 5-year zerocoupon bonds (dashed curve), and for an equal-weighted 118-bond portfolio of 5-year zero-coupons, one issued by each of the 118 firms of the 118-issuers sample described in Section 3 (solid curve). To put in perspective the n-th moment Sharpe ratios implied by the model of Section 2 (Figure 2), as a benchmark consider the S&P500 index, often used as a proxy for systematic risk. The S&P500 index has historical The S&P500 index return data is from CRSP. For the calculations in this sub-section I use annual data for the period For risk-free rates, I use 1-year US Treasury rates obtained from the Federal Reserve Board website. 23

24 1 0.8 s(t,t;2) s(t,t;3) s(t,t;4) Figure 2: From top to bottom, n-th moment Sharpe ratios associated with variance (n = 2), skewness (n = 3), and kurtosis (n = 4), respectively, for holding periods of up to five years. Dashed curve: Average (over time and cross-sectionally) of annualized n-th moment Sharpe ratio on individual 5-year zero-coupon bonds for the 118-issuer sample. Solid curve: Annualized n-th moment Sharpe ratios for the 118-issuer equal-weighted portfolio. annualized n-th moment Sharpe ratios of 0.272, 0.441, and for n = 2, 3, and 4, respectively. For individual 5-year zero-coupon corporate bonds, the average over the 24

25 118-issuer sample of 2nd moment Sharpe ratios, implied by the estimated model is approximately equal to that of the S&P500 index. For highermoment risk measures the average of the corporate bond Sharpe ratios are estimated to be smaller in absolute magnitude than their S&P500 counterparts. For the equal-weighted portfolio of 5-year zero-coupon corporate bonds, however, the n-th moment Sharpe ratios implied by the estimated model are much larger than their S&P500 counterparts for holding periods close to the maturity of the bonds, and are at least as large for shorter holding periods. For the same portfolio of 118 bonds, the left-hand panel of Figure 3 shows the estimated annualized holding-period Sharpe ratios implied by the model of Section 2, at each point in time over the sample period. The right-hand panel in Figure 3 shows the average, over time, of the Sharpe ratios from the left panel Annualized Sharpe Ratio Date Holding Period Annualized Sharpe Ratio Holding Period Figure 3: Left panel: Historical, annualized, holding-period Sharpe ratios, for an equal-weighted portfolio of 118, 5-year zero-coupon bonds. Right panel: Historical average of annualized, holding-period Sharpe ratios, for an equal-weighted portfolio of 118, 5-year zero-coupon bonds. One can attain even larger portfolio Sharpe ratios than those shown in Figure 3 by choosing portfolio weights, for each starting date and each holding-period, that maximize estimated Sharpe ratios. Figure 4 plots the Sharpe ratios implied by the model of Section 2 for a portfolio of 118 bonds, 25

26 with non-negative weights chosen to maximize the corresponding 2nd moment holding-period Sharpe ratio. The weights are constrained to be at most 0.1, ruling out short-sales and producing portfolios of at least 10 bonds Annualized Sharpe Ratio Date Holding Period Annualized Sharpe Ratio Holding Period Figure 4: Left panel: Historical, annualized, holding-period Sharpe ratios, for an optimally weighted portfolio of 118, 5-year zero-coupon bonds. The non-negative portfolio weights are chosen, for each starting date and each holding-period, to maximize the estimated portfolio Sharpe ratio, subject to a maximum weight of 10%. Right panel: Historical average of the annualized, holding-period Sharpe ratios presented in the left panel. The significant down-side risk and limited up-side potential of corporate bond returns is analogous to the risk of writing out-of-the-money put options. These generate small positive excess returns most of the time, and occasionally produce a large negative payoff. This suggests a comparison of the Sharpe ratios of the 118-name portfolio of corporate bonds to the corresponding 118-name portfolio of short out-of-the-money put options (on the same underlying names). Because of the scarcity of individual-name out-ofthe-money put option data, I instead use out-of-the-money put option data on the S&P500 index. An option on a portfolio, however, is much different than a portfolio of options. Figure 5 presents smooth-spline estimates of annualized n-th moment Sharpe ratios for writing 3-month (solid line) and 1-year (dashed line) put options on the S&P500 index, as a function of the underlying-to-strike ratio. 26

27 s(t,t;2) s(t,t;3) s(t,t;4) S/K Figure 5: Smooth-spline estimates of 2nd, 3rd, and 4th moment Sharpe ratios (from top to bottom) on short out-of-the-money put options on the S&P500 index as a function of the ratio of underlying to strike. The third- and fourth-moment Sharpe ratios for 3-month 20%-out-of-themoney put options on the S&P500 index are comparable to their counterparts for the 118-name portfolio of corporate bonds. Furthermore, while the 2nd-moment Sharpe ratio of the index is significantly smaller than that of the equal-weighted 118-name portfolio of corporate bonds, the 2nd-moment Sharpe ratio for writing 3-month 20%-out-of-the-money put options on the index is larger than that of the corporate bond portfolio at short holding periods, and smaller at long holding periods. However, the Sharpe ratios for the optimally-weighted portfolio (Figure 4) are much larger than those of writing 3-month and 1-year out-of-the-money put options on the index. The skewness and kurtosis of returns for writing 3-month 20%-out-ofthe-money put options on the index are, however, much larger than those 27

28 of the equal-weighted portfolio of corporate bonds (Figure 6). On the other hand, the skewness and kurtosis of returns for writing 1-year 20%-out-ofthe-money put options on the index are comparable to those of the equalweighted portfolio of corporate bonds, and thus their Sharpe ratios might provide a more relevant reference to which to compare the Sharpe ratios of the portfolio of corporate bonds. Figure 5 shows that the n-th moment Sharpe ratio for writing 1-year 20%-out-of-the-money put options on the index are much smaller than those of the equal-weighted portfolio of 5-year zero-coupon corporate bonds on 118 names Skewness Kurtosis Holding period S/K Figure 6: Left-hand plots: Estimates of skewness and kurtosis of returns for holding periods of up to 5 years on the equal-weighted 118-name portfolio of 5- year zero-coupon corporate bonds. Right-hand plots: Estimates of skewness and kurtosis for writing 3-month (solid line) and 1-year (dashed line) put options on the S&P500 index as a function of the underlying-to-strike ratio. Das, Duffie, Kapadia, and Saita (2006) test and reject, at standard confidence levels, the joint hypothesis of correctly specified default intensities and the doubly-stochastic assumption for the model of actual default probabilities of Duffie, Saita, and Wang (2005), on which I based the model of default probabilities for this paper. It is possible that the large 2nd moment Sharpe ratios for the equal-weighted portfolio of corporate bonds are due to a fail- 28